Linear Regression

• Factors influencing Life Expectancy using Linear Regression

• 1. Overview of the data
• 1. Data Visualization
• 1. Data Preparation
• 1. Choose Model, Train and Evaluate
• 1. Conclusion
• 1. Add-on: Statsmodels

Import Libraries

# Import necessary libraries
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

# To enable plotting graphs in Jupyter notebook
%matplotlib inline

import warnings
warnings.filterwarnings('ignore')

Load and explore the dataset

cdata  = pd.read_csv('/home/jayanthikishore/Downloads/ML_classwork/Week3strt/Life Expectancy Data.csv') 
cdata.head()

	Country	Year	Status	Life expectancy	Adult Mortality	Infant deaths	Alcohol	Percentage expenditure	Hepatitis B	Measles	...	Polio	Total expenditure	Diphtheria	HIV/AIDS	GDP	Population	Thinness 1-19 years	Thinness 5-9 years	Income composition of resources	Schooling
0	Afghanistan	2015	Developing	65.0	263.0	62	0.01	71.279624	65.0	1154	...	6.0	8.16	65.0	0.1	584.259210	33736494.0	17.2	17.3	0.479	10.1
1	Afghanistan	2014	Developing	59.9	271.0	64	0.01	73.523582	62.0	492	...	58.0	8.18	62.0	0.1	612.696514	327582.0	17.5	17.5	0.476	10.0
2	Afghanistan	2013	Developing	59.9	268.0	66	0.01	73.219243	64.0	430	...	62.0	8.13	64.0	0.1	631.744976	31731688.0	17.7	17.7	0.470	9.9
3	Afghanistan	2012	Developing	59.5	272.0	69	0.01	78.184215	67.0	2787	...	67.0	8.52	67.0	0.1	669.959000	3696958.0	17.9	18.0	0.463	9.8
4	Afghanistan	2011	Developing	59.2	275.0	71	0.01	7.097109	68.0	3013	...	68.0	7.87	68.0	0.1	63.537231	2978599.0	18.2	18.2	0.454	9.5

5 rows × 22 columns

cdata.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 2938 entries, 0 to 2937
Data columns (total 22 columns):
 #   Column                           Non-Null Count  Dtype  
---  ------                           --------------  -----  
 0   Country                          2938 non-null   object 
 1   Year                             2938 non-null   int64  
 2   Status                           2938 non-null   object 
 3   Life expectancy                  2928 non-null   float64
 4   Adult Mortality                  2928 non-null   float64
 5   Infant deaths                    2938 non-null   int64  
 6   Alcohol                          2744 non-null   float64
 7   Percentage expenditure           2938 non-null   float64
 8   Hepatitis B                      2385 non-null   float64
 9   Measles                          2938 non-null   int64  
 10  BMI                              2904 non-null   float64
 11  Under-five deaths                2938 non-null   int64  
 12  Polio                            2919 non-null   float64
 13  Total expenditure                2712 non-null   float64
 14  Diphtheria                       2919 non-null   float64
 15  HIV/AIDS                         2938 non-null   float64
 16  GDP                              2490 non-null   float64
 17  Population                       2286 non-null   float64
 18  Thinness  1-19 years             2904 non-null   float64
 19  Thinness 5-9 years               2904 non-null   float64
 20  Income composition of resources  2771 non-null   float64
 21  Schooling                        2775 non-null   float64
dtypes: float64(16), int64(4), object(2)
memory usage: 505.1+ KB

cdata.shape

(2938, 22)

The data-set aims to answer the following key questions:

• Do various predicting factors really affect the Life expectancy?
• What are the predicting variables actually affecting life expectancy?
• Should a country having a lower life expectancy value increase its healthcare expenditure in order to improve its average lifespan?
• Do infant and adult mortality rates affect life expectancy?
• Does life expectancy have positive or negative correlation with a country's status (developing or developed), lifestyle, GDP, etc.
• What is the impact of schooling on the lifespan of humans?
• Does life expectancy have a positive or negative relationship with drinking alcohol?
• What is the impact of immunization coverage (for various diseases like Measles,Hepatitis B) on life expectancy?

cdata.columns

Index(['Country', 'Year', 'Status', 'Life expectancy', 'Adult Mortality',
       'Infant deaths', 'Alcohol', 'Percentage expenditure', 'Hepatitis B',
       'Measles', 'BMI', 'Under-five deaths', 'Polio', 'Total expenditure',
       'Diphtheria', 'HIV/AIDS', 'GDP', 'Population', 'Thinness  1-19 years',
       'Thinness 5-9 years', 'Income composition of resources', 'Schooling'],
      dtype='object')

# remove the rows of data which have missing value(s)
cdata = cdata.dropna()

# Check the unique values in each column of the dataframe.
cdata.nunique()

Country                             133
Year                                 16
Status                                2
Life expectancy                     320
Adult Mortality                     369
Infant deaths                       165
Alcohol                             833
Percentage expenditure             1645
Hepatitis B                          83
Measles                             603
BMI                                 538
Under-five deaths                   199
Polio                                68
Total expenditure                   669
Diphtheria                           66
HIV/AIDS                            167
GDP                                1649
Population                         1647
Thinness  1-19 years                179
Thinness 5-9 years                  185
Income composition of resources     548
Schooling                           147
dtype: int64

Insights

• The "Status" column has 2 unique values. i.e. The values are "Developing" and "Developed"
• The "Country" column has 133 unique values. i.e. The data is collected from 133 countries.

Data Visualizations

plt.figure(figsize=(10,7))
plt.scatter(cdata['Schooling'], cdata['Life expectancy'], color='#c3a935')
plt.title('Life expectancy Vs Schooling', fontsize=14)
plt.xlabel('Schooling', fontsize=14)
plt.ylabel('Life expectancy', fontsize=14)
plt.grid(True)
plt.show()

plt.figure(figsize=(10,7))
plt.scatter(cdata['Measles'], cdata['Life expectancy'], color='#359dc3')
plt.title('Life expectancy Vs Measles ', fontsize=14)
plt.xlabel('Measles', fontsize=14)
plt.ylabel('Life expectancy', fontsize=14)
plt.grid(True)
plt.show()

plt.figure(figsize=(10,7))
plt.scatter(cdata['Adult Mortality'], cdata['Life expectancy'], color='#a035c3')
plt.title('Life expectancy Vs Adult Mortality', fontsize=14)
plt.xlabel('Adult Mortality', fontsize=14)
plt.ylabel('Life expectancy', fontsize=14)
plt.grid(True)
plt.show()

plt.figure(figsize=(10,7))
plt.scatter(cdata['Life expectancy'][:200], cdata['Country'][:200], color='#c36b35')
plt.title('Country Vs Life expectancy', fontsize=14)
plt.xlabel('Life expectancy', fontsize=14)
plt.ylabel('Country', fontsize=14)
plt.grid(True)
plt.show()

sns.pairplot(cdata, height=3, diag_kind='auto', corner=True)
plt.show()

• We can see from the 2nd and 3rd column of the graph that these features have linear (or inversely linear) relationship.

plt.figure(figsize=(10,10))
sns.boxplot(cdata['Life expectancy'], orient='v')
plt.show()

• The bottom black horizontal line of blue box plot is called 'minimum' and is calculated as Q1-1.5*IQR, which is around 48 years.
• First black horizontal line of rectangle shape of blue box plot is First quartile or 25 percentile. The age corresponding to this line is 64 years.
• Second black horizontal line of rectangle shape of blue box plot is Second quartile or 50% or median. The age corresponding to this is 72 years.
• Third black horizontal line of rectangle shape of blue box plot is third quartile or 75%. The age corresponding to this is 75 years.
• Top black horizontal line of rectangle shape of blue box plot is called 'maximum' and is calculated as Q3+1.5*IQR. The value corresponding to this is 88 years.
• Small diamond shape of blue box plot is outlier data or erroneous data. Which is below the bottom black horizontal line.

plt.figure(figsize=(8,8))
sns.boxplot(x="Status",y="Life expectancy",data=cdata)
plt.show()

• The life expectancy in the developed countries is much higher than the developing countries.
• The median value of life expectancy (approximately as can be seen from the boxplot) for:
  • Developing: 69 years
  • Developed: 78 years

cdata[cdata.columns[:]].corr()['Life expectancy'][:]

Year                               0.050771
Life expectancy                    1.000000
Adult Mortality                   -0.702523
Infant deaths                     -0.169074
Alcohol                            0.402718
Percentage expenditure             0.409631
Hepatitis B                        0.199935
Measles                           -0.068881
BMI                                0.542042
Under-five deaths                 -0.192265
Polio                              0.327294
Total expenditure                  0.174718
Diphtheria                         0.341331
HIV/AIDS                          -0.592236
GDP                                0.441322
Population                        -0.022305
Thinness  1-19 years              -0.457838
Thinness 5-9 years                -0.457508
Income composition of resources    0.721083
Schooling                          0.727630
Name: Life expectancy, dtype: float64

plt.figure(figsize=(20,20))
sns.heatmap(cdata.corr(), annot=True, fmt=".2")
plt.show()

Data Preparation

X = cdata.drop('Life expectancy', axis=1)
y = cdata[['Life expectancy']]

print(X.head())
print(y.head())

       Country  Year      Status  Adult Mortality  Infant deaths  Alcohol  \
0  Afghanistan  2015  Developing            263.0             62     0.01   
1  Afghanistan  2014  Developing            271.0             64     0.01   
2  Afghanistan  2013  Developing            268.0             66     0.01   
3  Afghanistan  2012  Developing            272.0             69     0.01   
4  Afghanistan  2011  Developing            275.0             71     0.01   

   Percentage expenditure  Hepatitis B  Measles   BMI  ...  Polio  \
0               71.279624         65.0     1154  19.1  ...    6.0   
1               73.523582         62.0      492  18.6  ...   58.0   
2               73.219243         64.0      430  18.1  ...   62.0   
3               78.184215         67.0     2787  17.6  ...   67.0   
4                7.097109         68.0     3013  17.2  ...   68.0   

   Total expenditure  Diphtheria  HIV/AIDS         GDP  Population  \
0               8.16        65.0       0.1  584.259210  33736494.0   
1               8.18        62.0       0.1  612.696514    327582.0   
2               8.13        64.0       0.1  631.744976  31731688.0   
3               8.52        67.0       0.1  669.959000   3696958.0   
4               7.87        68.0       0.1   63.537231   2978599.0   

   Thinness  1-19 years  Thinness 5-9 years  Income composition of resources  \
0                  17.2                17.3                            0.479   
1                  17.5                17.5                            0.476   
2                  17.7                17.7                            0.470   
3                  17.9                18.0                            0.463   
4                  18.2                18.2                            0.454   

   Schooling  
0       10.1  
1       10.0  
2        9.9  
3        9.8  
4        9.5  

[5 rows x 21 columns]
   Life expectancy
0             65.0
1             59.9
2             59.9
3             59.5
4             59.2

print(X.shape,y.shape )

(1649, 21) (1649, 1)

X = pd.get_dummies(X, columns=['Country', 'Status'])
X.head()

	Year	Adult Mortality	Infant deaths	Alcohol	Percentage expenditure	Hepatitis B	Measles	BMI	Under-five deaths	Polio	...	Country_Turkmenistan	Country_Uganda	Country_Ukraine	Country_Uruguay	Country_Uzbekistan	Country_Vanuatu	Country_Zambia	Country_Zimbabwe	Status_Developed	Status_Developing
0	2015	263.0	62	0.01	71.279624	65.0	1154	19.1	83	6.0	...	0	0	0	0	0	0	0	0	0	1
1	2014	271.0	64	0.01	73.523582	62.0	492	18.6	86	58.0	...	0	0	0	0	0	0	0	0	0	1
2	2013	268.0	66	0.01	73.219243	64.0	430	18.1	89	62.0	...	0	0	0	0	0	0	0	0	0	1
3	2012	272.0	69	0.01	78.184215	67.0	2787	17.6	93	67.0	...	0	0	0	0	0	0	0	0	0	1
4	2011	275.0	71	0.01	7.097109	68.0	3013	17.2	97	68.0	...	0	0	0	0	0	0	0	0	0	1

5 rows × 154 columns

Split the dataset

#split the data into train and test
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42)

Choose Model, Train and Evaluate

from sklearn.linear_model import LinearRegression
linearregression = LinearRegression()                                    
linearregression.fit(X_train, y_train)                                  

print("Intercept of the linear equation:", linearregression.intercept_) 
print("\nCOefficients of the equation are:", linearregression.coef_)

from sklearn.metrics import mean_absolute_error, mean_squared_error, r2_score

pred = linearregression.predict(X_test)

Intercept of the linear equation: [-370.39923249]

COefficients of the equation are: [[ 2.18341200e-01 -4.25965091e-04  7.28152544e-02 -1.15746797e-01
  -3.63099062e-05  1.99567177e-03 -4.41704129e-06 -5.08329484e-03
  -5.66462864e-02 -8.57849110e-04 -1.73216730e-02  2.20806233e-03
  -2.78374203e-01 -4.82283672e-07  1.96856187e-09  1.03678455e-02
   5.67478068e-02  5.30963077e-01  4.05983164e-01 -8.78321506e+00
   7.46457656e+00  4.92555587e+00 -1.42658382e+01  6.01520191e+00
   5.88048191e+00  1.94391020e+00  3.65381015e+00  2.32870321e+00
   1.41649354e+00  2.22867048e+00  1.59959045e+00  1.57736884e+00
  -8.60376699e+00 -2.24478634e+00  7.87205969e+00 -8.02143731e+00
   4.38559558e+00 -4.76806311e+00 -7.54778984e+00 -9.85626475e+00
   4.38903311e+00 -2.03678264e+00 -1.00404780e+01  1.34596802e+01
  -1.29481469e+01 -1.23459633e+01  1.06404340e+01  5.31590679e+00
   5.66500448e+00 -5.58832507e+00  1.08326967e+01 -1.71393790e+00
   1.51385537e+00 -3.68586682e+00  4.92769106e+00  6.64362218e+00
   3.66895438e+00  6.21724894e-15 -5.00185384e+00  5.82498645e+00
  -1.99677849e+00 -1.40945544e-01  1.40774573e+01 -2.91411032e+00
   6.03624515e+00  2.37740546e+00 -4.06255033e+00  1.29686571e+01
   7.13966581e+00 -8.81099198e+00 -8.98352872e+00 -1.09935918e+00
  -4.69095455e+00  5.94636052e+00 -1.19806439e+00 -1.05711230e+00
   4.15411611e-01  2.39181278e+00  1.19669073e+01  3.75763791e+00
   6.65903405e+00  4.32301087e+00 -1.87671319e+00 -7.36094991e+00
  -2.30040589e+00 -4.97686150e+00  5.62765944e+00 -1.25610871e+01
  -6.50097116e+00 -5.42286487e+00  3.29944724e+00 -3.46521938e+00
  -1.22923569e+01  4.84409505e+00  6.92397336e+00 -9.19859022e+00
   2.84022035e+00 -3.24066207e+00  3.71690721e+00  7.72158487e+00
  -2.22980311e+00  5.84847666e+00  4.53910570e+00 -9.69720382e+00
  -2.98729348e+00 -2.77460078e+00 -1.58338035e+00  9.42439431e-01
   5.47594821e+00 -2.33864142e+00 -4.41127302e+00 -3.25284580e+00
   8.75573924e+00 -4.08709466e+00  4.98511381e+00  5.42573096e+00
  -2.41166938e-01 -2.79062495e+00  1.35647266e+00 -5.49115792e+00
  -2.31232015e-02 -5.22956619e+00  5.68579389e+00 -1.75611274e+00
  -2.13942249e+00  6.11010903e+00  4.39779152e+00 -1.88261721e+01
   1.28478274e+00 -5.42805953e+00  2.24505808e+00  3.37555362e+00
   3.24079796e+00 -6.54386423e+00  2.27728958e+00  6.27853088e+00
  -9.79822596e-01  4.73178236e+00 -2.60229669e+00 -9.78893048e+00
   3.76723698e+00  3.54054284e+00  5.28635744e+00  4.71282558e+00
  -2.56002810e+00 -7.17678331e+00  2.20702234e+00  7.24732074e+00
  -1.13919257e-01  3.85522462e+00 -7.67892834e+00 -1.04447138e+01
   5.03543939e+00 -5.03543939e+00]]

Mean Absolute Error (MAE)

# Mean Absolute Error
mean_absolute_error(y_test, pred)

1.224371976020462

Root Mean Square Error (RMSE)

mean_squared_error(y_test, pred)**0.5

2.07571589731342

R Squared

# R2 Squared:
r2_score(y_test, pred)

0.9454598210791567

Training Score

# Training Score
linearregression.score(X_train, y_train)

0.9723226823336295

Testing Score

#Testing score
linearregression.score(X_test, y_test)

0.9454598210791567

Now, let's compare the actual output values for X_test with the predicted values by executing the following script:

predict = pd.DataFrame({'Actual': y_test.values.flatten(), 'Predicted': pred.flatten()})
predict

	Actual	Predicted
0	67.5	67.680534
1	73.8	73.785661
2	79.1	80.160082
3	54.9	53.138555
4	48.6	51.046567
...	...	...
490	64.8	65.959530
491	71.4	72.459268
492	77.2	77.069198
493	78.6	77.934373
494	68.0	68.095903

495 rows × 2 columns

• We can also visualize comparison result as a bar graph using the below script :

• Note: As the number of records is huge, for representation purposes we'll consider just 25 records.

df1 = predict.head(25)
df1.plot(kind='bar',figsize=(16,10))
plt.grid(which='major', linestyle='-', linewidth='0.5', color='green')
plt.grid(which='minor', linestyle=':', linewidth='0.5', color='black')
plt.show()

• We can observe here that our model has returned pretty good prediction results.
• The Actual and Predicted values are comparable.

Conclusion

• The Training and testing scores are around 95% and both scores are comparable, hence the model is a good fit.

• R2_score is 0.945, that explains 94.5% % of total variation in the dataset. So, overall the model performance is very satisfactory.

Add-on: statsmodels

import statsmodels.api as sm

X = sm.add_constant(X)

linearmodel = sm.OLS(y, X).fit()

predictions = linearmodel.predict(X) 

print_model = linearmodel.summary()
print(print_model)

                            OLS Regression Results                            
==============================================================================
Dep. Variable:        Life expectancy   R-squared:                       0.967
Model:                            OLS   Adj. R-squared:                  0.964
Method:                 Least Squares   F-statistic:                     294.9
Date:                Tue, 20 Apr 2021   Prob (F-statistic):               0.00
Time:                        15:07:59   Log-Likelihood:                -3100.3
No. Observations:                1649   AIC:                             6505.
Df Residuals:                    1497   BIC:                             7327.
Df Model:                         151                                         
Covariance Type:            nonrobust                                         
====================================================================================================
                                       coef    std err          t      P>|t|      [0.025      0.975]
----------------------------------------------------------------------------------------------------
const                             -248.7325     22.406    -11.101      0.000    -292.683    -204.782
Year                                 0.2219      0.017     12.837      0.000       0.188       0.256
Adult Mortality                     -0.0006      0.001     -1.194      0.233      -0.002       0.000
Infant deaths                        0.0497      0.016      3.204      0.001       0.019       0.080
Alcohol                             -0.0652      0.030     -2.151      0.032      -0.125      -0.006
Percentage expenditure           -7.639e-05      0.000     -0.625      0.532      -0.000       0.000
Hepatitis B                          0.0032      0.002      1.315      0.189      -0.002       0.008
Measles                          -6.555e-06   6.46e-06     -1.014      0.311   -1.92e-05    6.12e-06
BMI                                 -0.0015      0.003     -0.432      0.666      -0.008       0.005
Under-five deaths                   -0.0372      0.011     -3.402      0.001      -0.059      -0.016
Polio                               -0.0006      0.003     -0.231      0.817      -0.006       0.004
Total expenditure                   -0.0221      0.026     -0.837      0.403      -0.074       0.030
Diphtheria                           0.0009      0.003      0.300      0.764      -0.005       0.007
HIV/AIDS                            -0.3029      0.016    -19.157      0.000      -0.334      -0.272
GDP                                1.42e-05   1.85e-05      0.770      0.442    -2.2e-05    5.04e-05
Population                       -2.969e-10   9.29e-10     -0.320      0.749   -2.12e-09    1.52e-09
Thinness  1-19 years                 0.0115      0.033      0.351      0.726      -0.053       0.076
Thinness 5-9 years                   0.0664      0.031      2.124      0.034       0.005       0.128
Income composition of resources      0.9846      0.594      1.656      0.098      -0.181       2.151
Schooling                            0.2849      0.078      3.653      0.000       0.132       0.438
Country_Afghanistan                -10.1219      0.638    -15.870      0.000     -11.373      -8.871
Country_Albania                      6.4265      0.454     14.145      0.000       5.535       7.318
Country_Algeria                      4.1914      0.525      7.980      0.000       3.161       5.222
Country_Angola                     -16.5790      0.699    -23.728      0.000     -17.950     -15.208
Country_Argentina                    5.3187      0.627      8.478      0.000       4.088       6.549
Country_Armenia                      4.7513      0.454     10.456      0.000       3.860       5.643
Country_Australia                   -4.3413      0.634     -6.847      0.000      -5.585      -3.098
Country_Austria                     -3.3310      0.728     -4.575      0.000      -4.759      -1.903
Country_Azerbaijan                   2.2798      0.489      4.660      0.000       1.320       3.239
Country_Bangladesh                   0.8201      0.687      1.194      0.233      -0.528       2.168
Country_Belarus                      1.0368      0.588      1.763      0.078      -0.117       2.190
Country_Belgium                     -4.4922      0.669     -6.711      0.000      -5.805      -3.179
Country_Belize                       0.5589      0.456      1.226      0.221      -0.336       1.453
Country_Benin                       -9.8262      0.552    -17.794      0.000     -10.909      -8.743
Country_Bhutan                      -2.9737      0.662     -4.490      0.000      -4.273      -1.675
Country_Bosnia and Herzegovina       7.0048      0.538     13.011      0.000       5.949       8.061
Country_Botswana                    -8.0765      0.519    -15.569      0.000      -9.094      -7.059
Country_Brazil                       3.7303      0.537      6.944      0.000       2.676       4.784
Country_Bulgaria                   -11.5049      0.820    -14.039      0.000     -13.112      -9.897
Country_Burkina Faso                -9.4831      0.777    -12.202      0.000     -11.008      -7.959
Country_Burundi                    -11.1037      0.632    -17.569      0.000     -12.343      -9.864
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==============================================================================
Omnibus:                      667.315   Durbin-Watson:                   1.239
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             3360.711
Skew:                           1.861   Prob(JB):                         0.00
Kurtosis:                       8.921   Cond. No.                     1.42e+24
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The smallest eigenvalue is 4.22e-30. This might indicate that there are
strong multicollinearity problems or that the design matrix is singular.

Interpreting the Regression Results:

1. Adjusted. R-squared: It reflects the fit of the model.
   • R-squared values range from 0 to 1, where a higher value generally indicates a better fit, assuming certain conditions are met.
   • In our case, the value for Adj. R-squared is 0.964, which is good!
   • Note the difference in R^2 values from the sklearn model and OLS model. This is because we trained the sklearn model on training data and calculated the score using testing data but in statsmodels, we trained the model using the whole dataset and calculated the score on the same data.
2. const coefficient is the Y-intercept.
   • It means that if all the dependent variables (features: like Country, status, Adult mortality and, etc.) coefficients are zero, then the expected output for the dependent variable would be equal to the const coefficient.
   • In our case, the value for const coeff is -248.7325
3. Status coeff: It represents the change in the output Y due to a change in the status (everything else held constant).
4. Schooling coeff: It represents the change in the output Y due to a change of one unit in the Schooling (everything else held constant).
5. std err: It reflects the level of accuracy of the coefficients.
   • The lower it is, the higher is the level of accuracy.
6. P >|t|: It is p-value.
   • A p-value of less than 0.05 is considered to be statistically significant.
7. Confidence Interval: It represents the range in which our coefficients are likely to fall (with a likelihood of 95%).

You can read about other terms in the summary here.

Residual = Actual - Predicted

# Plot between residual(actual - predicted) and predicted values
plt.figure(figsize=(10,8))
plt.scatter(linearmodel.predict(), linearmodel.resid, marker='*')
plt.show()

Error Distribution

# error distribution
plt.figure(figsize=(10,8))
sns.distplot(linearmodel.resid, hist=True, kde=False, color='red')
plt.show()